| The Severi-Lefschetz Theorem implies that on smooth hypersurfaces in P^n, n large enough, the only complete intersection codimension 1 subvarieties are complete intersections of the surface with a hypersurface of another degree. We investigate the existence of complete intersection subvarieties of codimension bigger than 1 and show how the study of these subvarieties (their existence or not) is related to the Froberg conjecture in commutative algebra. The connection is formed by showing how that conjecture is related to questions about secants and joins of varieties of reducible forms. This is joint work with Enrico Carlini and Luca Chiantini. |